These threads are also useful, in addition to the main one that Raven linked:
The browser itself is called NetFront, and is made by ACCESS Co
from: https://browsedns.net/topic/25/br
And:
CaptiveDNS allows you to effectively run the same DNS redirection service as BrowseDNS, but on your local computer
from: https://browsedns.net/topic/2072/r
BrowseDNS provides the redirect / landing page, but the actual web browser is truly built into the Switch, and built in a partnership with Nintendo. ACCESS also made the WiiU and 3DS browsers, so it makes sense that they provided this for the Switch as well.
The Switch also uses the same browser in several places on the OS, such as for the eShop or the News and Share applets.